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ft1ln06_08 - MIT OpenCourseWare http://ocw.mit.edu 8.323...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 8.323 Relativistic Quantum Field Theory I Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.323: Relativistic Quantum Field Theory I Prof. Alan Guth April 2, 2008 LECTURE NOTES 6 PATH INTEGRALS, GREENS FUNCTIONS, AND GENERATING FUNCTIONALS In these notes we will extend the path integral methods discussed in Lecture Notes 5 to describe Greens functions, which we define to be ground state expecta- tion values of the time-ordered product of Heisenberg operators. For the case of a nonrelativistic particle moving in one dimension, discussed in Lecture Notes 5, the Greens functions can be written as G ( t N , . . . , t 1 ) | T { x ( t N ) x ( t N 1 ) . . . x ( t 1 ) }| (6.1) = | x ( t N ) x ( t N 1 ) . . . x ( t 1 ) | , where | denotes the ground state, and the second line assumes that we have labeled the time arguments so that they are time-ordered, in the sense that t N t N 1 . . . t 1 . (6.2) In the quantum field theory, the Greens functions will be defined analogously by G ( x N , . . . , x 1 ) | T { ( x N ) . . . ( x 1 ) }| , (6.3) = | ( x N ) . . . ( x 1 ) | , where | denotes the vacuum state, and again the second line assumes that the time arguments are time-ordered. In the nonrelativistic quantum mechanics example of Eq. (6.1), the Greens functions are not quantities that are particularly interesting, so they are usually never mentioned in a course in quantum mechanics. We will soon see, however, that the quantum field theory Greens functions of Eq. (6.3) are very interesting. In particular, the entire formalism for calculating scattering cross sections and decay rates will be based upon relating these quantities to the Greens functions. In addition to showing how to express these Greens functions as path integrals, in these notes we will also see that one can define a generating functional Z [ J ] in such a way that the Greens functions can be expressed simply in terms of the functional derivatives of the generating functional. 8.323 LECTURE NOTES 6, SPRING 2008 p. 2 Path Integrals, Greens Functions, and Generating Functionals GREENS FUNCTIONS: To begin, we recall that in Lecture Notes 5 we learned to express the evolution operator of quantum mechanics as a path integral: x ( t f )= x f U fi = D x ( t ) e i S [ x ( t )] , h (6.4) x (0)= x i where t f S [ x ( t )] = dt L ( x, x ) ( L = Lagrangian) (6.5) t f 1 = dt mx 2 V ( x ) . 2 We also know that the Heisenberg field operators appearing in Eq. (6.1) can be written as x ( t ) = e iHt x (0) e iHt , (6.6) where x (0) x is the Schr odinger representation position operator. Eq. (6.1) can S then be rewritten as G ( t N , . . . , t 1 ) = e iHt N x S e iH ( t N t N 1 ) x S . . . e iH ( t 2 t 1 ) x S...
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This note was uploaded on 11/08/2011 for the course PHY 8.323 taught by Professor Staff during the Spring '08 term at MIT.

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ft1ln06_08 - MIT OpenCourseWare http://ocw.mit.edu 8.323...

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